It is best not to think of the overhead press as a “shoulder press” because when done well it is a full body lift. By taking the attitude that you are about to perform a full body lift you will be more likely to cultivate full body tension. Full body tension demands that one start with their feet and either grip the floor or corkscrew the feet outwards without moving the feet themselves. This begins the ground connection and the chain of tension which will be taken up into the glutes, the belly, the lats, the upper back, and then completed by squeezing the bar with the hands.
Elbows behind the bar
For those who have the requisite flexibility, try and begin each rep with the elbows just slightly forward of the bar as it rests near the collarbone.This can help with lat tension and also will help with keeping the bar close to the face. Keep the bar close to the face and ensure that the bar travels in a path which ends locked out over the base of the neck. When elbows drift behind the bar during set-up or at the initiation of the press then it is much more likely that the bar will then drift away from the face and the body, thereby making the lift much more difficult to lockout and hold overhead.
Not putting the head through
For those with the flexibility, it is easier to lock the bar out and stabilize it by slightly bringing the head forward as the bar passes the face. At the end point of the movement the bar comes to it’s final stop above the base of the neck. Pressing this way has several advantages. It can take stress off the low back. It allows the skeletal structure to support the bar while at the same time allowing the traps to support more of the weight. Take care to not shove the head through too much thus putting the bar behind the neck and stressing it and the shoulder joint.
Too much use of stretch reflex
There is a time to use the stretch reflex, for example when doing some kind of challenge for number of reps. If the goal is to get strong, however, then a strict approach, pausing the bar in the bottom for each rep as one refills their breathe and resets their body tension, is best.
Too much knee bend
Keep the legs straight, unless deliberately doing a push press. Pay close attention to this on heavy sets and final reps. If you are doing a strict press then be strict.